Sharp
estimates for mean curvature flow of graphs are shown - a gradient estimate and
an area estimate - and examples are given to illustrate why these are sharp.
The gradient estimate improves an earlier (non-sharp) estimate of Klaus Ecker and Gerhard Huisken (joint
with Colding, Crelles
Journal, volume 574, 2004); LANL
link.

Given a Riemannian metric
on a homotopy $n$-sphere, sweep it out by a
continuous one-parameter family of closed curves starting and ending at point
curves. Pull the sweepout tight by, in a continuous
way, pulling each curve as tight as possible yet preserving the sweepout. We show:
Each curve in the tightened sweepout
whose length is close to the length of the longest curve in the sweepout must itself be close to a closed geodesic.
In particular, there are curves in the sweepout that
are close to closed geodesics.
Finding closed geodesics on the 2-sphere by using sweepouts goes back to Birkhoff
in 1917. As an application, we bound from above, by a negative constant,
the rate of change of the width for a one-parameter family of convex hypersurfaces that flows by mean curvature. The width is
loosely speaking up to a constant the square of the length of the shortest
closed curve needed to ``pull over'' $M$. This estimate is sharp and leads to a
sharp estimate for the extinction time; cf. above where a similar bound for the
rate of change for the two dimensional width is shown for homotopy
3-spheres evolving by the Ricci flow (see also Perelman).

This is an expository
article with complete proofs intended for a general non-specialist audience.
The results are two-fold. First, we discuss a geometric invariant, that we call
the width, of a manifold and show how it can be realized as the sum of areas of
minimal 2-spheres. For instance, when $M$ is a homotopy
3-sphere, the width is loosely speaking the area of the smallest 2-sphere
needed to ``pull over'' $M$. Second, we use this to conclude that Hamilton's
Ricci flow becomes extinct in finite time on any homotopy
3-sphere. We have chosen to write this since the results and ideas given here
are quite useful and seem to be of interest to a wide audience.

We prove a smooth
compactness theorem for the space of embedded self-shrinkers
in $\RR^3$. Since self-shrinkers model singularities
in mean curvature flow, this theorem can be thought of as a compactness result
for the space of all singularities and it plays an important role in studying
generic mean curvature flow.

It has long been
conjectured that starting at a generic smooth closed embedded surface in R^3,
the mean curvature flow remains smooth until it arrives at a singularity in a
neighborhood of which the flow looks like concentric spheres or cylinders. That
is, the only singularities of a generic flow are spherical or cylindrical. We
will address this conjecture here and in a sequel. The higher dimensional case
will be addressed elsewhere.

The key in showing this
conjecture is to show that shrinking spheres, cylinders and planes are the only
stable self-shrinkers under the mean curvature flow.
We prove this here in all dimensions. An easy consequence of this is that every
other singularity than spheres and cylinders can be perturbed away.

We show that for a Schrodinger operator
with bounded potential on a manifold with cylindrical ends the space of
solutions which grows at most exponentially at infinity is finite dimensional
and, for a dense set of potentials (or, equivalently for a surface, for a fixed
potential and a dense set of metrics), the constant function zero is the only
solution that vanishes at infinity. Clearly, for general potentials there can
be many solutions that vanish at infinity. These results follow from a three
circles inequality (or log convexity inequality) for the Sobolev
norm of a solution to a Schrodinger equation on a product $N\times [0,T]$,
where $N$ is a closed manifold with a certain spectral gap. Examples of such $N$'s are all (round) spheres $\SS^n$
for $n\geq 1$ and all Zoll
surfaces. Finally, we discuss some examples arising in geometry of such
manifolds and Schrodinger operators.

Minimal surfaces with
uniform curvature (or area) bounds have been well understood and the regularity
theory is complete, yet essentially nothing was known without such bounds. We
discuss here the theory of embedded (i.e., without self-intersections) minimal
surfaces in Euclidean 3-space without a priori bounds. The study is divided
into three cases, depending on the topology of the surface. Case one is where
the surface is a disk, in case two the surface is a planar domain (genus zero),
and the third case is that of finite (non-zero) genus. The complete
understanding of the disk case is applied in both cases two and three.
As we will see, the helicoid, which is a double
spiral staircase, is the most important example of an embedded minimal disk. In
fact, we will see that every such disk is either a graph of a function or part
of a double spiral staircase. The helicoid was
discovered to be a minimal surface by Meusnier in
1776.
For planar domains the fundamental examples are the catenoid,
also discovered by Meusnier in 1776, and the Riemann
examples discovered by Riemann in the beginning of the 1860s. Finally, for
general fixed genus an important example is the recent example by
Hoffman-Weber-Wolf of a genus one helicoid.
In the last section we discuss why embedded minimal surfaces are automatically
proper. This was known as the Calabi-Yau conjectures
for embedded surfaces. For immersed surfaces there are counter-examples by
Jorge-Xavier and Nadirashvili.

The study of embedded
minimal surfaces in $\RR^3$ is a classical problem, dating to the mid 1700's,
and many people have made key contributions. We will survey a few recent
advances, focusing on joint work with Tobias H. Colding
of MIT and Courant, and taking the opportunity to focus on results that have
not been highlighted elsewhere; LANL link.

This paper is the fifth and
final in a series on embedded minimal surfaces. Following our earlier papers on
disks, we prove here two main structure theorems for non-simply connected embedded minimal surfaces of any given fixed
genus. (joint with Colding);
LANL link.

We give a quick tour
through the field of minimal submanifolds. Starting
at the definition and the classical results and ending up with current areas of
research. Many references are given for further readings (joint with Colding; Bulletin of the London Math. Society);
LANL link.

In this paper we will prove
the Calabi-Yau conjectures for embedded surfaces. In
fact, we will prove considerably more. The Calabi-Yau
conjectures about surfaces date back to the 1960s. Much work has been
done on them over the past four decades. In particular, examples of Jorge-Xavier
from 1980 and Nadirashvili from 1996 showed that the
immersed versions were false; we will show here that for embedded surfaces,
i.e., injective immersions, they are in fact true. (Joint with Colding; LANL
link. Annals of
Mathematics 2008)

We construct a sequence of
(compact) embedded minimal disks in a ball where the curvature blows up only at
the center. This converges to a limit which is not smooth and not proper
(joint with Colding, Trans.
AMS, 2004); LANL
link.

Survey of Embedded minimal
surfaces I, II, and IV - intended also as a reader's guide (joint with Colding, The Proceedings of the Clay Mathematics Institute
Summer School on the Global Theory of Minimal Surfaces); LANL link.

Proves estimates on area
and total curvature for intrinsic balls in two-sided stable minimal surfaces in
three-manifolds; as consequences, we get Bernstein theorems and curvature
estimates. In the case of area, this curvature estimate is due to Schoen. (joint with Colding; IMRN 2002).